Computer Science + Math

I am considering doing a double major in computer science and math. What better compliments computer science, pure math or applied math (most likely with an emphasis on physics)? If it depends, then on what? Is proof-writing really useful for computer science?

Proof writing will come in handy if your designing new algorithms and have to formally prove the best case/worst case/average case etc

Depends on what your future goals are, grad school or not, software engineer or researcher, etc

Is it difficult to become a researcher in CS? That's what I'm primarily interested in. I thought academic jobs were difficult to get but the Department of Labor (http://www.bls.gov/oco/ocos042.htm) seems to say it is exactly the opposite for a computer scientist, which is relieving, if it is true.

I think both pure and applied math complement computer science. Pure math is better for theoretical computer science, that is automata theory, computability theory, as all subfields of theory of computation. As the previous post says, it's also very useful for algorithm proving.
Applied mathematics is very important for scientific computing. Topics like numerical analysis, finite element analysis, numerically solving differential equations, numerical simulations, all are topics covered both by computer science and applied mathematics.
It depends on what you prefer.

I think both pure and applied math complement computer science. Pure math is better for theoretical computer science, that is automata theory, computability theory, as all subfields of theory of computation. As the previous post says, it's also very useful for algorithm proving.
Applied mathematics is very important for scientific computing. Topics like numerical analysis, finite element analysis, numerically solving differential equations, numerical simulations, all are topics covered both by computer science and applied mathematics.
It depends on what you prefer.

I prefer pure math but quantum computing is really interesting and applied math I think is more appropriate for that (please correct me if I am wrong on this). But it is definitely good to know that both are helpful, thank you.

I am still confused though about the job prospects for a theoretical computer scientist; the Labor Department considers the 'computer scientist' group separate from software engineers and programmers, so is it safe to assume that those "excellent job prospects" are in research?

Quantum computing is very theoretical and therefore I think that pure mathematics will help you more than applied mathematics, if you want to pursue quantum computing. Of course, a lot of physics is also a pre.
As for the job prospects, I think most of the theoretical computer scientists end up as researchers for the simple reason that I don't think there are any jobs in industry for theoretical computer scientists. Of course, one can study theoretical computer science at uni and then work in the industry as a software engineer for example. But there aren't many jobs (if any) in which their knowledge of theoretical computer science is directly applied. Most people actually see theoretical computer science as a subfield of pure mathematics. It's no coincidence that most theoretical computer scientists are mathematicians by education. In industry, pure mathematicians are mainly hired for their analytical and intellectual skills and not for their knowledge, in say for example, algebraic topology. For theoretical computer scientists it's the same situation.
On the other hand, I think that computer science along with applied mathematics has a lot of applications in real world, and therefore in industry. Think about computer graphics, computational geometry (which is very important for computer aided design), numerical algorithms (which is very important for every field of engineering). Those are very important topics which have many applications in all fields of engineering and science but found there basis in computer science and whose theories are still mainly developed by computer scientists. Personally I find these topics more interesting than theoretical computer science, mainly because they have applications in real world.
Of course, what you want to do is your own choice. I hope this helps.

Quantum computing is very theoretical and therefore I think that pure mathematics will help you more than applied mathematics, if you want to pursue quantum computing. Of course, a lot of physics is also a pre.
As for the job prospects, I think most of the theoretical computer scientists end up as researchers for the simple reason that I don't think there are any jobs in industry for theoretical computer scientists.

So I could just major in CS + pure math and take some quantum physics courses on the side? Would that be better than doing CS + applied math (for physics)? If so, then great, I'm set, thanks a lot for the help. I still can't believe how great the job prospects are for a CS researcher!

Such topics as machine learning, pattern recognition, neural networks are based heavily on mathematics. If I understand correctly, the researchers of these fields should be proficient in mathematics ? They must be mathematicians, not the computer scientists. The math education of comp science undergraduates is not equal in comparison with math majors. What do you think ?

All computer scientists should know about mathematics. For many fields of computer science though, they need to know different mathematics than a typical mathematics major would need to know. Typical areas computer scientists need to know a lot about are discrete mathematics (graph theory, recurrence relations, etc...), probability, algebra... There is a subtle difference between computer scientists and mathematicians though. Computer scientists use the mathematics to "formalize" their ideas: to them, mathematics is just a tool to describe their ideas and algorithms. In that way, there are just like physicists. For mathematicians, mathematics is a goal per se. They have no other goal beyond it.
So yes, computer scientists need to be proficient in mathematics but not as much as mathematicians. Because it's seen more as a tool, the mathematics needed for a certain subject is often seen "on the way" during the course.
The wikipedia page on computer science gives a very clear view of all the fields of computer science: http://en.wikipedia.org/wiki/Computer_science" [Broken]